12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Efficient Optimal Design-Under-Uncertainty of Passive StructuralControl DevicesSubhayan DeGraduate Research Assistant, Sonny Astani Department of Civil and EnvironmentalEngineering, University of Southern California, Los Angeles, California, USASteven F. WojtkiewiczAssociate Professor, Department of Civil and Environmental Engineering,Clarkson University, Potsdam, New York, USAErik A. JohnsonProfessor, Sonny Astani Department of Civil and Environmental Engineering,University of Southern California, Los Angeles, California, USAABSTRACT: This paper proposes a computationally efficient framework for design optimization underuncertainty for structures with local nonlinearities. To reduce the high computational cost of MonteCarlo simulation of such problems, an exact model reduction to a low-order Volterra integral equationis used to accelerate each simulation, and variance-reduced sampling is used to reduce the number ofsimulations required for the uncertainty quantification. This optimization framework is applied to abenchmark cable-stayed bridge problem, designing one pair of passive tuned mass dampers given a pairof uncertain passive power law dampers, providing significant gains in computational efficiency, twoorders of magnitude, compared to traditional approaches.1. INTRODUCTIONRobust design in presence of uncertainties in ma-terial, loading or topological characteristics of astructure has been investigated using convex op-timization, neural network and evolutionary algo-rithms (Enevoldsen and Sørensen, 1994; Sandgrenand Cameron, 2002; Papadrakakis and Lagaros,2002; Zang et al., 2005; Calafiore and Dabbene,2008). Over the past few decades, reliability-based design optimization of these uncertain struc-tures has been investigated using single and multi-objective optimization both approaches (Frangopol,1985; Gasser and Schuëller, 1997; Tu et al., 1999;Adeli, 2002; Beck et al., 1996, 1999).While many design-level earthquakes generatesuperstructure responses that are elastic and linear,passive structural control devices embedded in thestructure often have nonlinear characteristics (e.g.,power-law, bilinear and/or hysteretic behavior), in-troducing local nonlinearities into an otherwise lin-ear model and, therefore, requiring either simplify-ing approximations or a full nonlinear simulationto evaluate response characteristics. If some partsof such a model are also uncertain (e.g., character-istics of the same, or other, structural control ele-ments or other localized components), then MonteCarlo sampling can be used for uncertainty charac-terization; this, in turn, further increases the compu-tational requirements for analysis of such systems.The optimal design of passive nonlinear control de-vices requires repeated solutions of the nonlinearsystem. Multiplying these three costs — nonlin-ear simulation, Monte Carlo sampling, and designparameter iterations — often creates a significantcomputational burden, so there is a clear advantagein the design-under-uncertainty problem for a com-putationally efficient approach to solve for the re-sponse of locally nonlinear systems.A conventional nonlinear solver (e.g., ode45 inMATLAB) cannot exploit the localized nature of the112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015nonlinearities. However, a method recently pro-posed by the last two authors (Gaurav et al., 2011)develops an exact nonlinear model reduction forsystems of this type, resulting in a low-order non-linear Volterra integral equation, providing a signif-icant computational speedup compared to ode45.Further, the authors recently proposed (Kamalzareet al., 2015) capitalizing on the local nature of de-sign variables to make design optimization simula-tions more computationally efficient.This paper extends the scope of that work by ad-ditionally incorporating localized uncertainties intothe model to arrive at a computationally-efficientdesign-under-uncertainty framework for systemsthat are mostly linear and deterministic but thathave localized nonlinear and uncertain elements.The proposed framework is illustrated using a fi-nite element model of the Bill Emerson MemorialBridge in Cape Girardeau, Missouri, consisting of579 nodes, 162 beam elements, 128 cable elements,420 rigid links and 134 nodal masses, resultingin a 419 degree-of-freedom (DOF) model after re-moving dependent or boundary DOFs (Dyke et al.,2003). The optimal design of parameters of lin-ear and nonlinear structural control devices is per-formed using the proposed approach while key el-ements of the structure are uncertain (e.g., a previ-ously installed passive damping device). The pro-posed method is shown to provide significant re-duction in required computation times, relative toode45, while delivering the same level of accuracy.2. METHODOLOGY2.1. Efficient response with local uncertaintiesFollowing Gaurav et al. (2011), let the nonlinearstructure model be given in state space byX˙(t) = AX(t)+Bw(t)+Lugu(Xu(t);δ )+Ldgd(Xd(t);θ )= AX(t)+Bw(t)+Lg(X(t);δ ,θ )Y(t) = CX(t)+Dw(t)+Eugu(Xu(t);δ )+Edgd(Xd(t);θ )= CX(t)+Dw(t)+Eg(X(t);δ ,θ )X(0) = x0(1)where X(t) is the n×1 state vector; A is then×n state matrix, w is an m×1 external excita-tion; B is the n×m influence matrix; gu(·; ·) isan ng,u × 1 function of a subset Xu(t) = GuX(t)of states and uncertain parameters δ ; gd(·; ·) is anng,d × 1 function of a subset Xd(t) = GdX(t) ofstates and design parameters θ ; Lu is an n×ng,uinfluence matrix mapping to all states from theforce vector gu arising due to uncertain parame-ters; Ld is an n×ng,d influence matrix mappingto all states from the force vector gd(·; ·) arisingdue to design parameters; D is an ny×m influencematrix; Eu and Ed are ny×ng,u and ny×ng,d in-fluence matrices, respectively; and x0 is the ini-tial condition. State subsets Xu(t) and Xd(t) areno,u×1 and no,d×1 vectors, respectively, withno,u,no,d n. Output Y(t) is an ny×1 vec-tor. Combining the forces from the uncertaintiesand nonlinearities, L = [LTu LTd ]T is n×no andg(X(t);δ ,θ ) = [gTu (Xu(t);δ ) gTd (Xd(t);θ )]T isno×1, where no ≤ (ng,u + ng,d). Similarly, X(t) =[XTu (t) XTd (t)]T = GX(t) for G = [GTu GTd ]T.The nominal linear system corresponding to thenonlinear system in (1) isx˙(t) = Ax(t)+Bw(t), x(0) = x0 (2)Using the principle of superposition, the responseof the system can be divided into two parts: the so-lution x(t) of the nominal linear system in (2) andthe contribution of the functions g(X) which can bewrittenx(t) =eAtx0 +∫ t0HB(t− s)w(s)ds,x(nl)(t) =∫ t0HL(t− s)g(X(s);δ ,θ )ds(3)where the impulse responses are given by HB(t) =eAtB and HL(t) = eAtL. The total response is thesuperposition of these two responses; i.e., X = x+x(nl). For any value of δ and θ , the solution of (3)can be computed efficiently with the following:p(t) = g(X(t);δ ,θ );X(t) = x(t)+∫ t0HL(t− s)p(s)ds(4)where x(t) = Gx(t) and HL(t) = GHL(t). The setof equations (4) can be combined into:p(t)−g(x+∫ t0HL(t− s)p(s)ds;δ ,θ)= 0 (5)212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Equation (5) is a nonlinear vector Volterra inte-gral equation (NVIE) written in nonstandard form,which can be solved by a Newton-Gregory integra-tion scheme (Linz, 1985). In the algorithm pro-posed in the authors’ earlier works, the Newton-Gregory integration is coupled with a recursive FFTformulation to drastically reduce the cost of compu-tation of the required convolution.Of course, any redundant columns in L (e.g., iftwo devices are collocated) can be eliminated andthe dimension of g(X(t);δ ,θ ) can be reduced, sim-plifying the solution of NVIE (5). Further, any re-dundant columns of G can be eliminated and thedimensions of x and X can be reduced.2.2. Design optimization under uncertaintyTwo design-under-uncertainty objectives are em-ployed herein: worst-case design and average de-sign (Calafiore and Dabbene, 2008). A brief de-scription of these two procedures follows.2.2.1. Worst case designIn this design method, the structure or control de-vice is designed for the case when the cost functionis maximized over the domain of uncertainty withconstraints satisfied, known as worst-case design.This design optimization problem can be defined asminθ∈Θmaxδ∈∆J(Y(t);θ ,δ )subject to h(Y(t);θ ,δ max(θ ))≤ 0 a.s.1(6)where J(·) and h(·) may be functionals of the en-tire trajectory of Y; the set of all possible values ofdesign parameter θ is denoted by Θ; ∆ representsa probability space {Ω,P,F} with sample spaceΩ, probability measure P, and σ -algebra F corre-sponding to the uncertainty defined for the problem;δ max(θ ) ∈ ∆ is the uncertainty realization which,for a particular design θ , maximizes J(Y(t);θ ,δ )subject to the constraint h; and J(·) is assumed con-cave in δ . The analytical solution of (6) may not al-ways be possible. However, with samples {δ i}Nδi=1from ∆, one may approximate the problem asminθ∈Θmaxi=1,...,NδJ(Y(t);θ ,δ i)subject to h(Y(t);θ ,δ i,max(θ ))≤ 0(7)1Almost sure (a.s.) event happens with probability 1.where δ i,max(θ ) corresponds to the sample which,for a particular design θ , maximizes J(Y(t);θ ,δ )subject to the constraint h.2.2.2. Average designIn the second method of design considered, the ex-pected value of cost is minimized while keeping theconstraints satisfied. This optimization problem isdefined asminθ∈ΘEδ [J(Y(t);θ ,δ )]subject to h(Y(t);θ ,δ )≤ 0 a.s.(8)With samples {δ i}Nδi=1 from ∆,minθ∈Θ1NδNδ∑i=1J(Y(t);θ ,δ i)subject to h(Y(t);θ ,δ i)≤ 0(9)3. NUMERICAL EXAMPLE3.1. Cable-stayed bridge modelThe example used to demonstrate the framework isa numerical model of the Bill Emerson MemorialBridge, a cable stayed bridge, built in 2003 acrossthe Mississippi river between Cape Girardeau, Mis-souri, and East Cape Girardeau, Illinois. A finiteelement model of the bridge superstructure, devel-oped in Dyke et al. (2003) and shown in Fig. 1, con-sists of 579 nodes, 128 cable elements, 162 beamelements, 420 rigid links and 134 nodal masses.(The version of the model used herein has no con-nection between deck and tower except throughthe cables so as to allow energy dissipator devicesplaced between the deck and a tower.) The initial3474 degree-of-freedom (DOF) model, which de-scribes the superstructure’s linear motion about thestatic equilibrium, is reduced to 909 DOFs whenthe boundary conditions are imposed and the slaveDOFs removed. Static condensation is then ap-plied to eliminate DOFs with small contribution tothe global response, resulting in the final 419 DOFmodel, which is available publicly (Dyke et al.,2003) and which is used here. The equation of mo-tion of this final bridge model is given byMsu¨s(t)+Csu˙s(t)+Ksus(t) =−Msru¨g(t) (10)312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Figure 1: Finite element model of the bridge (dimensions in m); adapted from Dyke et al. (2003).where Ms, Cs, and Ks are the mass, damping andstiffness matrices, respectively, of the bridge super-structure for the active DOFs; r is the influence vec-tor for the ground acceleration (containing ones inentries corresponding to active displacements in thedirection of the 1-D horizontal excitation and zeroselsewhere); us(t) is the generalized displacementvector; and u¨g(t) is the ground acceleration in thelongitudinal direction. The reader is directed to thebenchmark definition paper (Dyke et al., 2003) forfurther details of this model.3.2. Passive damping devicesTwo types of passive devices are investigated: qunonlinear viscous dampers and qd nonlinear tunedmass dampers (TMDs). The damping forces in theviscous dampers follow the power-law relationf ui = cui |∆u˙i|β ui sgn(∆u˙i), i = 1, . . . ,qu (11)where ∆u˙i is the velocity across the ith damper. Thegoverning differential equations of the TMD massmotions are given by,mdi v¨i + cdi |∆˙vi|β di sgn(∆˙vi)+ kdi ∆vi=−mdi rdi u¨g, i = 1, . . . ,qd(12)where vi is the displacement of the TMD relativeto the ground, ∆vi is the TMD displacement rela-tive to its attachment point on the bridge, and rdi isin [−1,1] depending on the TMD’s orientation rel-ative to the earthquake ground motion direction.To be consistent with the symmetric nature ofthe bridge model, the passive devices are placedin identical pairs symmetrically located aboutthe bridge deck centerline; i.e., let cu2i−1 = cu2iand β u2i−1 = β u2i for i = 1, . . . ,qu (dampers) andcd2i−1 = cd2i, β d2i−1 = β d2i, md2i−1 = md2i and rd2i−1 = rd2ifor i= 1, . . . ,qd (TMDs). Further, this example con-siders each pair of nonlinear viscous dampers as un-certain — δ =[cu2 β u2 | cu4 β u4 | . . . | cuqu βuqu]T— and the stiffness and damping of each pairof TMDs as the design parameters — θ =[cd2 β d2 kd2 | cd4 β d4 kd4 | . . . | cdqd βdqd kdqd]T.The ratio of TMD masses relative to bridge mass isfixed at a certain value.3.3. FormulationThe bridge equations of motion can be written inform of (1) where,X(t) ={u(t)u˙(t)}, A =[0 I−M−1K −M−1C],B ={0r¯}, Lu =[0−M−1Ru], Ld =[0−M−1Rd],u(t) ={us(t)v(t)}, M=[Ms 00 Md], K=[Ks 00 0],and C =[Cs 00 0]; Md is a diagonal matrix ofTMD masses; r¯ = [rT r1 . . .rqd]T is the influence412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015vector for the ground acceleration; Ru and Rd arethe influence matrices for the uncertain damperforces and the TMD forces (from design parame-ters), respectively. Each of the qu columns of Ruand qd columns of Rd transform the pseudoforce ofone passive device to a global force vector, withgu(Xu(t);δ )=−cu2|∆u˙1|β u2 sgn(∆u˙1)−cu2|∆u˙2|β u2 sgn(∆u˙2)...− cuqu|∆u˙qu−1|β uqu sgn(∆u˙qu−1)−cuqu|∆u˙qu|β uqu sgn(∆u˙qu),where Xu(t)=[∆u˙1 ∆u˙2 | . . . | ∆u˙qu−1 ∆u˙qu]T=GuX(t) =[0 RTu]X(t) andgd(Xd(t);θ ) =−cd2|∆v˙1|β d2 sgn(∆v˙1)−kd1∆v˙1−cd2|∆v˙2|β d2 sgn(∆v˙2)−kd2∆v˙2...− cdqd|∆v˙qd−1|β dqd sgn(∆v˙qd−1)−kdqd∆v˙qd−1−cdqd|∆v˙qd|β dqd sgn(∆v˙qd)−kdqd∆v˙qdwhere Xd(t)= [∆v1 ∆v˙1 ∆v2 ∆v˙2 | . . . | ∆vqd−1∆v˙qd−1 ∆vqd ∆v˙qd]T = GdX(t) = [0 RTd ]X(t).3.4. Objective function and constraintsThe goal of this example is to optimize the param-eters of the TMDs to improve the bridge’s perfor-mance given uncertainties in the nonlinear viscousdampers installed elsewhere in the bridge. Differentperformance metrics can be chosen; a set of met-rics, normalized with respect to the uncontrolledand connected deck-tower case suggested by Dykeet al. (2003), are used here subject to the 1940 ElCentro earthquake excitation. As in Dyke et al.(2003): Fbi(t) and Mbi(t) are the base shear andoverturning moment, respectively, at the ith towerat time t; Fdi(t) and Mdi(t) are the correspondingdeck-level shear and overturning moment; || · (t)||denotes the root mean square (i.e., two-norm overtime) response; (·)max0(·) denotes the maximum, overboth time and tower number, uncontrolled base ordeck-level shear or overturning moment at the baseor deck level; and ||(·)0(·)(t)|| denotes the maxi-mum, over tower number, time-normed uncontrol-lable shear or overturning moment at the base ordeck level. Then, eight of the metrics used to formthe objective function and some constraints are,J1 =maxi,t|Fbi(t)|Fmax0b, J2 =maxi,t|Fdi(t)|Fmax0d,J7 =maxi||Fbi(t)||||F0b(t)||, J8 =maxi||Fdi(t)||||F0d(t)||J3 =maxi|Mbi(t)|Mmax0b, J4 =maxi|Mdi(t)|Mmax0d,J9 =maxi,t||Mbi(t)||||M0b(t)||, J10 =maxi,t||Mdi(t)||||M0d(t)||(13)Additional metrics used in the constraints areJ6 = maxi,t∣∣∣∣xbi(t)x0b∣∣∣∣, J12 = maxi,tfi(t)W, (14)where xbi(t) is the displacement of the deck at thetwo ends (Bent 1 and Pier 4) in the finite elementmodel at time t; x0b is the maximum, over time andend locations, of the uncontrolled displacement ofthe deck; fi(t) is the amount of force exerted bythe ith device, assumed here to be the TMD; andW = 510 MN is the weight of the bridge super-structure. The normed responses are calculated us-ing 200 s of response as suggested by Dyke et al.(2003). (Note: performance metrics J5, J11 and J13through J18, defined in the benchmark (Dyke et al.,2003), are not used herein.)Using these performance metrics, the determin-istic optimization problem is formulated asminθ∈ΘJ1(θ ) (15)subject to Jk(θ )≤ αJk,0 ∀k ∈ {2–4,7–10}where Θ = {θ : klb ≤ θ3 j−2 ≤ kub,clb ≤ θ3 j−1 ≤cub,β lb ≤ θ3 j ≤ β ub, j = 1, . . . ,qd}; Jk,0 cor-responds to the kth performance metric withoutdampers (i.e. cdi = 0, kdi = 0); (·)lb and (·)ub arelower and upper bounds on the design parameters;and α is set to 1.25 (De et al., 2015). Each TMD isdesigned with a mass that is 2% of the bridge mass;i.e., mdi = 0.02W/g for i = 1, . . . ,qd.512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015DamperlocationsTMD locationsFigure 2: Finite element model (side view) of thebridge; adapted from Dyke et al. (2003).For simplicity, consider the case of a single pairof TMDs attached on top of the 2nd tower as shownin Figure 2. The initial values k0 and c0 of thedesign parameters for the optimization problemare taken from the guidelines specified in Hoanget al. (2008) for the first mode of vibration withβ0 = 1.0 (clearly, any optimization should evalu-ate the effect of different initial values; here, withdifferent initial design points, the objective func-tion minimum was found to be relatively insensitivethough the optimum design point may change). Thelower and upper bounds for stiffness and dampingcoefficients are taken as [50 kN/m, 5 MN/m] and[100 kN·(s/m)β , 6 MN·(s/m)β ], respectively, afterpreliminary studies indicated that the optimum pa-rameters are expected to lie in these ranges. Thelower and upper limits for β are taken as 0.2 and1.8 as suggested by Main and Jones (2002).This optimization problem is then modified toinclude the design under uncertainty formulation.The uncertain parameter vector corresponds to asingle pair of nonlinear dampers (i.e., qu = 2), oneplaced connecting nodes 319 and 186 in the finiteelement model and the other symmetrically con-necting nodes 324 and 119 (see Figure 2). Theuncertain parameters of the passive dampers fol-low the distributions given in Table 1, where themean values were obtained from a deterministicoptimization similar to (15) performed over a setTable 1: Uncertain parameter descriptionVariable Distribution Mean COVDamping Log- 20.5440 0.05coeff.(cu2) normal MN·(s/m)β u2Exponent Log- 0.9777 0.1(β u2 ) normalNote: COV = coefficient of variationof possible passive viscous damper locations andparameters with c ∈ [0.5,30] MN·(s/m)β and β ∈[0.2,1.8].3.5. Optimization ResultsThe optimization is performed using MATLAB’sfmincon with default values for tolerances but lim-ited to 50 function evaluations (actual results mayslightly exceed 50 since each iteration may performmultiple function evaluations). An active-set algo-rithm (sequential quadratic programming) is usedinside fmincon. While fmincon can be providedwith gradient information, a finite difference ap-proximation is used here to evaluate the gradients.A preliminary study of this problem showed thatNδ = 100 samples, using Latin Hypercube sam-pling method with antithetic variates variance re-duction (Ayyub and Lai, 1991), were sufficient togive converged results for the average design op-timization (though the number of samples clearlydepends on the application and the optimization ob-jective and constraints). The results of the designoptimization are shown in Tables 2 and 3. The op-timal values for kd2 are similar for both average andworst case design; however the optimal values forTable 2: Worst-case design of nonlinear TMDReduction of # fcn. evals.cost: max J1 (in %)26.4922 51kd2 cd2 β d2[MN/m] [MN · (s/m)βd2 ]0.9086 3.8165 0.2000Table 3: Average design of nonlinear TMDReduction of # fcn. evals.cost: E [J1] (in %)7.3690 53kd2 cd2 β d2[MN/m] [MN · (s/m)βd2 ]0.8453 1.9186 1.0826612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20150 10 20 30 40 50 60−0.2−0.15−0.1−0.0500.050.10.150.2Time (s)Displacement (m) ode45 NVIE, ∆t = 0.46 ms(a) Accuracy in a typical generalized displacement0 20 40 60 80 100 120 140 160 180 20005101520253035Time (s)Cumulative sum of squared displacement (b) Cumulative sum of a typical squared responseFigure 3: Accuracy of the NVIE approach for first 60 s compared to MATLAB’s ode45cd2 and β d2 differ significantly. In the worst case de-sign, the value of device force performance met-ric J12 increases to 3.19 times that of the averagedesign case. A deterministic optimization for theTMD parameters with mean values for the passivedampers gives kopt = 0.9503 MN/m, copt = 0.3896MN·(s/m)βopt , βopt = 0.7568, which gives a lowerpeak device force and 0.29 times J12 than averagedesign case. Hence, due to the assumed uncertaintyin the structure, the robust optimization requires aTMD capable of exerting greater force.3.6. Computational EfficiencyThe nonlinear Volterra integral equation approachinvolves a one-time cost and repeated costs for eachiteration of the design under uncertainty frame-work. The one-time computational cost involvessolution of x(t) in (3) and calculation of HL. Therepeated cost involves solution of (5) and for theresulting outputs Y(t) for different uncertainty anddesign parameters. To evaluate the computationalcost of the proposed method, it is compared witha fixed time step 4th order Runge-Kutta method(RK-4). The RK-4 method diverges for ∆t = 1 msor higher because of the numerical stiffness of thesystem of differential equations of motion. Hence,∆t = 0.5 ms is used for RK-4, giving a relative ac-curacy of O(10−3) computed relative to the ode45with relative tolerance of 10−6 and absolute tol-erance of 10−8 (for just the first 60 s of responsewhich provides RMS close to the long term asshown in Figure 3b; only 60 s is used because thefull ode45 solution takes too much time). The pro-posed NVIE approach gives relative accuracy inRMS of states in the same order with ∆t = 0.46 ms.For Nδ = 100, and for 10 function evaluations inthe optimization procedure, x(nl)(t) must be evalu-ated a total of 1000 times. Computation times areevaluated using MATLAB’s cputime function on acomputer with a 2.3 GHz Core i7-4850HQ proces-sor, 16 GB RAM, Mac OS X, and running MAT-LAB 2013a. The proposed NVIE approach takes13.53 cpu-min. to compute p(t), 2.16 cpu-min. tocompute Y(t) in each repeated calculation and atotal 10.92 cpu-days for 1000 simulations. How-ever RK-4 requires the full solution a total 1000times, where each simulation takes 77.66 cpu-hrs.,which projects to a total of 8.87 cpu-years for thefull optimization. Hence, the proposed design-under-uncertainty framework provides a computa-tional speedup of 296.33 with comparable accuracy.While only 3 design variables are considered here,comparable or increased gains in computational ef-ficiency, relative to other approaches for simulatingthe system responses for the function evaluations,are expected when more design variables are used.4. CONCLUSIONSStructural design under uncertainties providesmany computational challenges. The proposed712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015method shows significant computational advan-tages for such problems. In future work, the reli-ability based design optimization will be incorpo-rated in this framework; further, design optimiza-tion under uncertainty of structural systems, withpassive control devices, subjected to random exci-tation will be considered in subsequent studies.ACKNOWLEDGMENTSThe authors gratefully acknowledge the partial sup-port of this work by the National Science Foun-dation through awards CMMI 13-44937 and 14-36018. Any opinions, findings, and conclusionsor recommendations expressed in this material arethose of the authors and do not necessarily reflectthe views of the National Science Foundation. Theauthors also acknowledge the support of the first au-thor by the Provost’s Ph.D. Fellowship at the Uni-versity of Southern California.5. 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